Keeping Texture Mode Appearance While Exporting to Unity
To make your model while rendered have that bright red color you see in Texture shading 3D view mode you can setup it in 2 ways:


Make the material shadeless:



The result:



However, it's not good approach as it's much less flexible; you will loose possibility of the material to have shadows and thus it may not suit your needs.
You can setup a texture for your material.

With that material selected ("red full.001" in your case) go to Texture tab and create new one. Select it's type Blend and check Ramp option in  Colors scroll. Delete one of the color stops, select another and copy the color from your material setup to this color stop (it may be easier to use Hex triplet for that):


Note: the texture should be checked to influence on the material (near its name in the list); I forgot to check it back when I made screenshots

The resulting material is bright, at the same time receiving shadows:



I also checked Color checkbox in Specular options in the Influence scroll (so the texture to influence on the specularity, instead of material; to return material specularity color uncheck that).